Three-dimensional surface contouring using Moire techniques is well known (see for example U.S. Pat. No. 4,212,073 issued July 8, 1980 to Balasubramanian). There are two basic implementations of this technique: Moire topography and Moire deflectometry. Moire topography is used to measure surfaces that are diffuse reflectors. Moire deflectometry is used with surfaces that are specular reflectors. The type of data that is collected with either of these techniques is similar and requires similar processing.
An example of a system used for Moire topography is shown in FIG. 1. Parallel equispaced lines of light or fringes 10 having a period d are formed for example by projecting the image of a grating 18 onto a surface 12 to be measured. The scattered intensity is viewed at an angle .theta. from the direction of the fringe projection by a lens 14 and camera 16 having an image sensor 17. If the surface 12 is not flat, a curved fringe pattern is seen, and the curvature is related to the surface profile. The intensity pattern on the surface is given by: EQU i(x,y)=i'(x,y)+i"(x,y) cos [(2.pi./d)(x+h(x,y) tan .theta.)], (1)
where i'(x,y) and i"(x,y) are constants relating to the DC bias and AC modulation of the viewed fringes, d is the period of the projected fringes on the surface, h(x,y) is the surface height profile, and the fringes are projected parallel to the y-axis. Only the fundamental frequency component of the projected grating 18 is given by this equation--if the original grating is a square-wave target, for example a Ronchi ruling, higher frequency terms will appear in equation (1). The information about the surface height in the viewed fringe pattern is encoded in the spacing of the fringes. The surface height difference needed to increase or decrease the period of the viewed fringes by the base fringe period d is known as the contour interval C, where EQU C=d/tan .theta.. (2)
There are at least two method for producing the projected fringes. The first is an optical projector as shown in FIG. 1. The grating transparency 18 is illuminated by a light source 20 and is imaged onto the surface 12 by the lens 22. To eliminate magnification errors with depth of field, a telecentric lens system is often used. However, projectors which produce diverging fringe patterns are also used, and the analysis is more complicated. In that event, the contour interval C is not a constant and changes as a function of height and position. Parallel, equispaced fringes are assumed for this analysis. The second method for projecting fringes is to use an interferometer as shown in FIG. 2. The interferometer is in a Twyman-Green configuration. A beam of collimated laser light 26 is split into the two arms of the interferometer by the beamsplitter 28, and parallel equispaced fringes 30 are produced by interference when the two beams are recombined. The period of the fringes can be changed by tilting one of the two mirrors 32.
The fringe pattern is sampled by the image sensor 17, and the sampled values are supplied to a computer 33. The computer 33 processes the samples, and displays the processed samples on a display, such as a CRT 35. A standard keyboard 37 is employed to input information into the computer to control the Moire contouring apparatus.
The other implementation of Moire contouring is Moire deflectometry. An example of such a system is shown in FIG. 3. As diagramed, a collimated input beam 38 is reflected off a specularly reflecting surface 40 at an angle .alpha.. The reflected beam passes through a grating 42 (having a pitch d) and impinges on a detector 44 that is a distance A from the grating. A surface deformation 46 having a slope of .beta. will deflect the reflected beam through an angle of 2.beta.. A distorted shadow of the grating 42 falls on the detector 44. The displacement of the fringe at a given location will be a function of the local deviation angle .beta., and the displacement equals 2.beta.A. In this situation, the contour interval C.sub.D occurs when this displacement is equal to the grating period d: EQU C.sub.D =d/2A. (3)
Note that the contour interval is an angle, not a distance. In addition to the differences in the surface reflectivities, this condition marks the major difference between Moire topography and deflectometry. Moire deflectometry measures the slope of the surface perpendicular to the grating lines, while Moire topography gives the surface height distribution.
Without loss of generality, the majority of the remainder of the discussion will center on the application of Moire topography. It should be clear, however, that the principles described here are equally applicable to Moire deflectometry.
There are several schemes available for detecting and measuring the curvature of the viewed fringe pattern. One scheme involves viewing the pattern through a second grating, identical to the projection grating, that is located in the focal plane of the camera 16. A Moire or product of the distorted test pattern and the reference pattern is formed, hence the name Moire topography. When the grids are chosen to be identical, a flat object will produce a zero-frequency beat or difference frequency. The resulting Moire pattern will have no fringes in it after the high spatial frequencies corresponding to the individual gratings are filtered out. When the surface is not flat, low frequency Moire fringes, corresponding to the difference between the reference and test patterns, will appear in the filtered output. The Moire pattern is sampled, and the fringe centers are located. The surface profile can then be constructed by using the fact that adjacent Moire fringes indicate a change of the surface height of one contour interval C.
An often more convenient method of sampling the Moire pattern is to image the distorted test pattern directly onto a solid-state detector array. The spacing of the pixels themselves in the array will serve as the reference grid, and the phenomenon of aliasing will result in the low frequency Moire pattern, which is the product of the reference grating (i.e. the sensing array) and the deformed projected grating pattern. To simplify the analysis, all dimensions will be discussed as they appear in the sensor plane. For example, scale factors relating to the magnifications of the imaging systems will be ignored.
For the purpose of modeling the system, the sensor geometry diagramed in FIG. 4 is assumed. A rectangular array of rectangular pixels 48 is used, and the pixels have dimensions and spacings of a by b and x.sub.s by y.sub.s respectively. The sampled image i.sub.s (x,y) produced by the sensor is EQU i.sub.s (x,y)=[i(x,y)**rect(x/a, y/b)]comb(x/x.sub.s,y/y.sub.s), (4)
where i(x,y) is the viewed fringe pattern as described by equation (1), ** indicates a two-dimensional convolution, and the comb function is an array of delta functions with spacing of x.sub.s by y.sub.s. The frequency space representation of the sampled pattern is obtained by taking a Fourier transform: EQU I.sub.s (.xi.,.eta.)=[I(.xi.,.eta.) sinc(a.xi.,b.eta.)]**comb(x.sub.s .xi.,y.sub.s .eta.), (5)
where .xi. and .eta. are the spatial frequency coordinates, I(.xi.,.eta.) is the spectrum of the intensity pattern, and ##EQU1## The active area of the pixel, represented by the rect and sinc functions in equations (4) and (5), serves to reduce the contrast of the recorded fringe pattern. The intensity pattern is averaged over the pixel, and the contrast of a fringe at a particular spatial frequency or spacing is reduced by the corresponding value of the sinc function.
The nature of the previously mentioned aliasing and the limitations in measurement range of the conventional Moire techniques can be observed by plotting equation (5) in one dimension for two different types of viewed patterns. The product I(.xi.)sinc(a.xi.), which is the input scene, is shown in FIG. 5a. A sinusoidal fringe pattern is assumed, and this product contains three portions--the DC bias at zero frequency 50 and two lobes 52 centered around the frequencies .+-.1/d. These lobes contain the information about the curvature and spacing of the fringe pattern. Since a pattern of frequency 1/d is projected, one would expect the spectral content of the viewed pattern to be centered at these frequencies. The maximum width of these lobes is denoted W, and it is not necessary that W be centered on the frequency 1/d.
Before proceeding, it is useful to define a few terms. The Nyquist frequency f.sub.N of a sensor is defined to be half the sampling frequency of the sensor, or 1/2x.sub.s. This frequency is generally considered to be the limiting resolution of a sampled imaging system. The baseband of the sensor is the frequency range near the origin between .+-.f.sub.N. A Moire interval has been defined for this application as a frequency band of width f.sub.N that begins (and ends) at a multiple of the Nyquist frequency.
The first object to consider is one which is close to being flat. In this case, the width W is less than f.sub.N, and the proper choice of d will place the sidelobe of the viewed pattern in the second Moire interval, at frequencies between f.sub.N and 1/x.sub.s. The sampling operation replicates this spectrum at the multiples of the sampling frequency (equation (5)), and this situation is shown in FIG. 5b. All of the replicated lobes are well separated, and the first-order replicas 54 of the lobes have been mapped into the baseband of the sensor. The initial frequencies 56 in the viewed pattern, centered at 1/d, are beyond the Nyquist frequency of the sensor, and have been aliased to lower spatial frequencies. This effect subtracts the reference grid, the sensor sampling pattern, from the viewed fringe pattern. Note that a viewed fringe occurring at the same frequency as the samples is mapped back to zero frequency--the same result as when viewing the pattern through a reference grating. Since the orders are separated, the sampled spectrum can be easily low-pass filtered to yield only the desired information in the sensor baseband, the Moire fringes.
When the object is more steeply curved than that represented in FIG. 5b, the range of frequencies observed in the viewed intensity pattern will be greater, and W will increase. If W is greater than the Nyquist frequency, the sidelobe will no longer fit into a Nyquist interval, and the resultant spectrum of the sampled image is given in FIG. 5c. The lobes overlap and can no longer be separated. A measured fringe frequency in the baseband cannot be identified as having originated in just the first order side lobe 58 as required for the Moire analysis. It may also be present in the baseband as a result of the overlap of the other replicas 60. Prior art methods of Moire analysis have not been able to resolve this dilemma, and have failed to properly reconstruct objects of this type.
The maximum measurement range of conventional Moire topography can be calculated from the foregoing analysis. The maximum frequency difference in the viewed fringes must be less than the Nyquist frequency--this frequency range must fit into the baseband of the sensor. The maximum period of a Moire fringe must therefore be twice the sample pitch of the sensor or a half fringe per sample. Since a contour interval corresponds to a full fringe, the maximum surface change that is allowed by this analysis is a half of a contour interval per sample (C/2 per sample). This constraint limits the maximum slope of the surface that is being measured and indirectly limits the maximum excursion.
A method to obtain the relative phase of the fringe pattern at each sample is to use synchronous detection or phase-shifting techniques as disclosed in the referenced U.S. Pat. No. 4,212,073. According to this method, an arbitrary phase shift is inserted between the reference grid (the sensor) and the viewed intensity pattern by laterally shifting either the projection grating or the sensor. Mechanisms for obtaining this phase shift are easy to implement. In FIG. 1, the projection grating 18 can be translated by a transducer 24 in the direction of arrow B to obtain the shift. The interferometrically produced pattern in FIG. 2 can be phase shifted by translating either of the mirrors 32 in a direction perpendicular to its surface. A piezoelectric translater 34 is used for this purpose to move one of the mirrors in the direction of arrow C. These shifts can be computer controlled, with a time-varying intensity pattern being produced.
A general expression for the intensity pattern that is viewed by the detector 17 for a particularly phase shift .delta..sub.n is EQU i.sub.n (x,y)=i'(x,y)+i"(x,y) cos [(2.pi./d)(x+h(x,y) tan .theta.-.delta..sub.n)], (7)
where n indexes the phase shift, and the other terms are as described in equation (1). The data set needed for the analysis of the surface height using phase-shifting techniques is a set of three or more patterns produced with different phase shifts .delta..sub.n. A simple phase-shift algorithm to implement is the four-step method, where the phase is advanced in four equal steps of 90.degree.. In this case, EQU .delta..sub.n =0, d/4, d/2, 3d/4. (8)
The four recorded intensity patterns after trigonometric simplification are EQU i.sub.1 (x,y)=i'(x,y)+i"(x,y) cos [(2.pi./d)(x+h(x,y) tan .theta.)] EQU i.sub.2 (x,y)=i'(x,y)+i"(x,y) sin [(2.pi./d)(x+h(x,y) tan .theta.](9) EQU i.sub.3 (x,y)=i'(x,y)-i"(x,y) cos [(2.pi./d)(x+h(x,y) tan .theta.] EQU i.sub.4 (x,y)=i'(x,y)-i"(x,y) sin [(2.pi./d)(x+h(x,y) tan .theta.)].
Combining these equations and solving for h(x,y) gives the result ##EQU2## The last term in this equation is a linear tilt which corresponds to the projected uniform fringe pattern. This term is automatically removed by the Moire process or the aliasing that occurs with the solid-state sensor. Neither of these effects are included in this equation, but the term can be safety ignored. The term disappears when the reference grating is subtracted from the viewed fringe pattern. Noting also that d/tan .theta. is equal to the contour interval C, this equation can be rewritten as ##EQU3## This last equation is evaluated at every point on the surface to yield a map of the surface height. If the signs of the numerator and denominator of equation (11) are determined, the arctangent can be calculated over a range of 2.pi.. The result of equation (11) is therefore to assign a surface height to each sample that is a number between zero and C, the contour interval.
The next step in the analysis for Moire topography leads to its limitation. The result of the arctangent in equation (11) is to give the surface height modulo C. FIG. 6 is a plot of an actual surface profile 62 and the surface profile modulo C 64 as would result from the above calculation. In order for the data to be useful, the discontinuities resulting from the arctangent must be removed from the calculated profile 64. In other words, the calculated surface height 64 returns to a value of zero every time the actual surface height 62 equals a multiple of C, and this segmented or "compressed" surface must be resembled to obtain the correct result. The procedure that is used in Moire topography to remove these discontinuities is to start at a single sample of the surface contour, normally at the center of the surface, and to assume that the height change between any two samples is always less than C/2. If the surface height difference calculated for two adjacent samples exceeds C/2, then C is added to or subtracted from the value of the second sample until the above condition is met. The entire surface is then reconstructed by working outward in this manner for the starting sample. FIG. 7 illustrates this process, where the original calculated surface height 66 is indicated by crosses, and the reconstructed values 68 are noted by circles. A dashed line has been drawn through the reconstructed values.
As was noted above, for a proper reconstruction, the surface height must change by less than C/2 per sample. This method of measurement therefore places a restriction on the types of surfaces that can be correctly measured. The slope of the surface must be less than C/2 per sample, and only surfaces that have small departures from a flat can be tested. For highly curved surfaces, the surface height changes too rapidly for the reconstruction algorithm to keep up with it. Because of the arctangent, this constraint in reconstruction is often referred to as the 2.pi. ambiguity problem.
Attempts by others to overcome this limitation have basically fallen into two categories. First, the number of samples can be increased to widen the Moire intervals, or second, the projected fringe frequency can be decreased to increase the contour level. However, since the resolution of these systems is a fraction of a contour interval, often between C/100 and C/1000, this latter change will also decrease the absolute resolution of the system. The resolution of these systems is signal-to-noise ratio dependent and is dictated by the errors in the arctangent calculation. Contributing factors include the number of bits in the digital computations and sensor noise.
Another detection scheme used in Moire topography is called spatial synchronous detection or spatial heterodyne. According to this technique, the viewed fringe pattern is recorded and digitized at high spatial resolution, and the reference grid is generated in the computer. The two patterns are then multiplied, and the samples of the Moire fringes are produced in the computer. The computer-generated grid can also be shifted to yield a phase-shifting technique similar to what has been described. The trade-off between this technique and the temporal phase-shifting approach is that more pixels are needed in the sensor. An advantage is that only one record of the test pattern is needed. The spatial heterodyne technique does not require a mechanism to shift the phase of a reference grating and is immune to temporal variations such as vibration. Range limitations, analogous to those described above, exist for the spatial heterodyne technique. Spatial heterodyne Moire topography is analogous to the phase-shifting technique, where all the information is sensed at once, and the samples of the Moire pattern are generated in a computer.
Accordingly, it is the object of the present invention to extend the useful range of Moire contouring past the Nyquist frequency limit of conventional Moire contouring without increasing the number of samples of the Moire pattern and/or without increasing the projected fringe frequency.